Nonlinear variation of diffusion exponents in quantum dynamics

We prove the existence of a nonlinear variation of the diffusion exponents in quantum dynamics. More precisely, we derive new lower bounds for the moments of order p associated to the state psi(t) = e(-itH)psi and averaged in time between 0 and T. These lower bounds are expressed in terms of generalized fractal dimensions D(1/(1 + p/d)) of the measure mu(psi) (where d is the space dimension). This improves notably previous results, obtained in terms of Hausdorff and Packing dimension. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.